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An abstract algebra problem!

What is the strongest condition you can up with for \(\alpha\) such that the field extension over \(F\) is equal to the ring of polynomials over \(F\). Or in other words \(F(\alpha)=F[\alpha]\)?

Note by Ali Caglayan
3 years, 5 months ago

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The strongest I could come up with is that \(\alpha\) generates a group such that \(\langle\alpha\rangle\cong C_n\) where \(C_n\) is the \(n^\text{th}\) cyclic group of arbitrary order.

Ali Caglayan - 3 years, 5 months ago

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I have gone further and said that \(\alpha\) has to be algebraic over \(F\).

Ali Caglayan - 3 years, 5 months ago

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